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(1)

Other polarimeters

• Wave plates can be used to build

polarimeters sensitive to linear or to circular polarization.

• Half-wave plate (φ=180

o

)

• Quarter-wave plate (φ=90

o

)

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

0 1

0 0

1 0

0 0

0 0

1 0

0 0

0 1 QWP

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= −

1 0

0 0

0 1

0 0

0 0

1 0

0 0

0 1 HWP

(2)

Polarimeter with HWP

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

0 0

0 0

0 0

0 0

0 0

1 1

0 0

1 1

PH

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= −

1 0

0 0

0 1 0

0

0 0

1 0

0 0

0 1 HWP

θ

source Polarizer

Intensity detector HWP

( ) ( )

{ P R HWP R } S

D

W = ×

H

× − θ × × θ ×

( )

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

1 0

0 0

0 0

0 0

0 0

0 1

2 2

2 2

c s

s R θ c

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= V U Q

I

S D=

(

1 0 0 0

) ( )

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

−

1 0

0 0

0 0

0 0

0 0

0 1

2 2

2 2

c s

s R θ c

(3)

Polarimeter with HWP

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

1 0

0 0

0 0

0

0 0

0

0 0

0 1

1 0

0 0

0 0

0 0

0 0

0 1

1 0

0 0

0 1 0

0

0 0

1 0

0 0

0 1

2 2

2 2

2 2

c c

c s

s c

θ

source Polarizer

Intensity detector HWP

( ) ( )

{ P R HWP R } S

D

W = ×

H

× − θ × × θ ×

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

⎟⎟ −

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

1 0

0 0

0 0

0

0 0

0

0 0

0 1

1 0

0 0

0 0

0

0 0

0

0 0

0 1

1 0

0 0

0 0

0 0

0 0

0 1

2 2 2

2 2

2 2

2

2 2

c c

c c

c s

s c

(4)

Polarimeter with HWP

θ

source Polarizer

Intensity detector HWP

( ) ( )

{ P R HWP R } S

D

W = ×

H

× − θ × × θ ×

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

⎟⎟ −

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

0 0 0 0

0 0 0 0

0 0 1

0 0 1

1 0

0 0

0 0

0

0 0

0

0 0

0 1

0 0 0 0

0 0 0 0

0 0 1 1

0 0 1 1

2 2 2 2

2 2 2

2 c

c

c c

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛ + +

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

0 0 0

0 0 0

0 0 0 0

0 0 1

0 0 1

2 2 2 2 2

2 2 2

Q c I

Q c I

V U Q

I c

c

( )

2

(

22

)

1 2

2 2 2

2 1

0 0 0

0 0

1 I c Q I Qc

Q c I

W = +

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛ + +

=

( )

f

o

f

Q I

W

4

2 cos

2

2 1

=

+

= θ

(5)

Polarimeter with QWP

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

=

0 0

0 0

0 0

0 0

0 0

1 1

0 0

1 1

PH

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

0 1

0 0

1 0

0 0

0 0

1 0

0 0

0 1 QWP

θ

source Polarizer

Intensity detector QWP

( ) ( )

{ P R QWP R } S

D

W = ×

H

× − θ × × θ ×

( )

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

1 0

0 0

0 0

0 0

0 0

0 1

2 2

2 2

c s

s R θ c

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= V U Q

I

S D=

(

1 0 0 0

) ( )

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

= −

−

1 0

0 0

0 0

0 0

0 0

0 1

2 2

2 2

c s

s R θ c

(6)

Polarimeter with QWP

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

0 0

1 0

0 0

0 0

0 0

0 1

1 0

0 0

0 0

0 0

0 0

0 1

0 1 0 0

1 0

0 0

0 0 1 0

0 0 0 1

2 2

2 2

2 2

2 2

c s

s c

c s

s c

θ

source Polarizer

Intensity detector QWP

( ) ( )

{ P R HWP R } S

D

W = ×

H

× − θ × × θ ×

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

−

= −

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

⎟⎟ −

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

0 0

0 0

0 0

0 0

0 1

0 0

1 0

0 0

0 0

0 0

0 1

1 0

0 0

0 0

0 0

0 0

0 1

2 2

2 2

2

2 2

2 2

2

2 2

2 2

2 2

2 2

c s

s c

s s

c c

c s

s c

c s

s c

(7)

Polarimeter with QWP

θ

source Polarizer

Intensity detector QWP

( ) ( )

{ P R HWP R } S

D

W = ×

H

× − θ × × θ ×

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

−

−

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

0 0

0 0

0 0

0 0 1 1

0 0

0 0

0 0

0 0

0 1

0 0 0 0

0 0 0 0

0 0 1 1

0 0 1 1

2 2

2 2

2

2 2

2 2

2

2 2

2 2

2

2 2

2 2

2 c c s s

s s

c c

c s

s c

s s

c c

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

− +

+

− +

+

=

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

0 0 0

0 0

0

0 0

0 0

1 1

2 2

2 2

2

2 2

2 2

2 2

2 2 2

2

2 2

2 2

2

V s U s c Q c I

V s U s c Q c I

V U Q

I s

s c c

s s

c c

( cos

2

2 θ cos 2 θ sin 2 θ sin 2 θ )

2

1

I Q U V

W = + + +

f

o

4 4 f

o

2 f

o

(8)

Modulation techniques

for polarization experiments

All these techniques suffer for the need of long integration time:

One needs to point to the same sky pixel during many cycles of

the analyzer. It is thus difficult to produce extended maps of the

CMB polarization.

Riferimenti

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